Finite derivation type implies the homological finiteness condition \(FP_ 3\)
Let \(M\) be a finitely generated monoid. A monoid presentation will consist of a finite alphabet \(\Sigma\) and a subset \(R\subseteq\Sigma^*\times\Sigma^*\) of rewriting rules (in the sense of formal grammar). If \(M\) is isomorphic to the monoid \(\Sigma^*\)/(the congruence generated by \(R\)), then the pair \((\Sigma;R)\) is called a presentation for \(M\). To say that a presentation \((\Sigma;R)\) is of finite derivation type means, roughly, that there is a finite set of derivations (in the sense of term rewriting) which, up to pre- and post-multiplication by elements of \(\Sigma^*\), account for all word equivalences in \(\Sigma^*\). Specifically, one first forms a graph \(\Gamma=\Gamma(\Sigma; R)\) with vertices \(V=\Sigma^*\) and, for \(u, v \in \Sigma^*\) and \((l,r) \in R\), edges \((u, (l, r), v, \pm 1)\) going from \(ulk\) to \(urv\) (in the case of \(+1\), and in the opposite direction in the case of \(-1\)). In this graph one can then consider the set of paths, \(P(\Gamma)\), and the equivalence relation \(P^{(2)}\) on these paths which has two paths \(p\) and \(q\) equivalent exactly when they have the same source and target vertices. Let \(B\) be a subset of \(P^{(2)}\) (considered as a set of pairs of paths). Then the homotopy relation generated by \(B\) on \(P(\Gamma)\) is the smallest relation \(\simeq\) which satisfies i. if \(p \simeq q\), then \(xpy \simeq xqy\) for all \(x, y \in \Sigma^*\), ii. if \(p \simeq q\) and \(r, s \in P(\Gamma)\), then \(r \circ p \circ s \simeq r \circ q \circ s\), provided the composition of paths is defined, and iii. \(D \cup I \cup B \subseteq \simeq\), where \(D\) represents the ``commuting squares'' resulting from applying two rewriting rules \((l_1, r_1)\) and \((l_2, r_2)\) in the two possible orders, and \(I\) represents the relation forcing a length 2 path consisting of traversing (essentially) the same edge ``out and back'' to be equivalent to the identity. If there is a finite \(B\) such that the homotopy relation generated by \(B\) is \(P^{(2)}\), then \((\Sigma; R)\) is of finite derivation type. The property of having a presentation of finite derivation type is a property of the monoid \(M\), in the sense that if one finite presentation for \(M\) has this property, then all do. Also, if \(R\) is a convergent rewriting system, then \(M\) will have this property. (The authors attribute this result to earlier work by C. C. Squire.) To investigate homological properties of a monoid \(M\) one forms the monoid ring \({\mathbf Z} M\) and considers a sequence of \({\mathbf Z} M\)- homomorphisms \(C_k \to \cdots \to C_2 \to C_1 \to C_0 \to {\mathbf Z} \to 0\) in which the kernel of each homomorphism is the image of the one to its left. Such a sequence is exact. The monoid \(M\) is said to satisfy condition \(FP_k\) if there is such a sequence with each \(C_n\) a finitely generated free \({\mathbf Z} M\)-module. The main result of this paper is \(\bullet\) If \(M\) is a finitely presented monoid having finite derivation type, then \(M\) satisfies the homological finiteness condition \(FP_3\). The proof uses the properties of the rewriting system to carry through the standard equational proof of exactness of the appropriate sequence. Some of the constructions involved seem (to this reviewer) related in spirit to earlier work by J. Leech for monoids and J. Duskin for groups, in which presentations by way of categories are used to construct certain resolutions. The authors go on to investigate the consequences of finite derivation type for groups. They present the following result \(\bullet\) A finitely generated group \(G\) has finite derivation type iff for every finite presentation \((\Sigma; L)\) of \(G\), the group \(N_0\) of relational identities associated with \((\Sigma; L)\) admits a finite defining system.
homological finiteness condition \(FP_ 3\)
scientific article; zbMATH DE number 696963